3.3.33 \(\int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx\)

Optimal. Leaf size=23 \[ \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{9 x^9} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \begin {gather*} \frac {\sqrt {x^6-1}}{9 x^9}+\frac {2 \sqrt {x^6-1}}{9 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^10*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(9*x^9) + (2*Sqrt[-1 + x^6])/(9*x^3)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2}{3} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2 \sqrt {-1+x^6}}{9 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^10*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(1 + 2*x^6))/(9*x^9)

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IntegrateAlgebraic [A]  time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^10*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(1 + 2*x^6))/(9*x^9)

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fricas [A]  time = 0.47, size = 26, normalized size = 1.13 \begin {gather*} \frac {2 \, x^{9} + {\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/9*(2*x^9 + (2*x^6 + 1)*sqrt(x^6 - 1))/x^9

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6-1)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [abs(x)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Val
ue

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maple [A]  time = 0.08, size = 20, normalized size = 0.87

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{9 x^{9}}\)	\(20\)
risch	\(\frac {2 x^{12}-x^{6}-1}{9 x^{9} \sqrt {x^{6}-1}}\)	\(25\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (2 x^{6}+1\right )}{9 x^{9} \sqrt {x^{6}-1}}\)	\(40\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (2 x^{6}+1\right ) \sqrt {-x^{6}+1}}{9 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^10/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*(x^6-1)^(1/2)*(2*x^6+1)/x^9

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maxima [A]  time = 0.33, size = 25, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^10/(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 - 1)/x^3 - 1/9*(x^6 - 1)^(3/2)/x^9

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mupad [B]  time = 0.29, size = 25, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x^6-1}+2\,x^6\,\sqrt {x^6-1}}{9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^10*(x^6 - 1)^(1/2)),x)

[Out]

((x^6 - 1)^(1/2) + 2*x^6*(x^6 - 1)^(1/2))/(9*x^9)

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sympy [A]  time = 0.93, size = 63, normalized size = 2.74 \begin {gather*} \begin {cases} \frac {2 \sqrt {x^{6} - 1}}{9 x^{3}} + \frac {\sqrt {x^{6} - 1}}{9 x^{9}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {2 i \sqrt {1 - x^{6}}}{9 x^{3}} + \frac {i \sqrt {1 - x^{6}}}{9 x^{9}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**10/(x**6-1)**(1/2),x)

[Out]

Piecewise((2*sqrt(x**6 - 1)/(9*x**3) + sqrt(x**6 - 1)/(9*x**9), Abs(x**6) > 1), (2*I*sqrt(1 - x**6)/(9*x**3) +
I*sqrt(1 - x**6)/(9*x**9), True))

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